Question

Solve the following equation $x^4+9x^3+12x^2-80x-192=0$, which has equal roots.

Answer 1

Given equation, $x^4+9x^3+12x^2-80x-192=0$

Consider $f\left(x\right)=x^4+9x^3+12x^2-80x-192$

$\therefore f^{\prime }\left(x\right)=4x^3+27x^2+24x-80$

Now, HCF of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is $(x+4)$. Hence $-4$ is atleast a double root of $f\left(x\right)=0$

$f\left(x\right)$ can be factored as $(x+4{)}^2(x^2+x-12\left)\text{or}\right(x+4{)}^3(x-3)$

$\therefore$ Roots of the given equation are $-4,-4,-4,3$